<h2>Problem 126</h2>
<div style="color:#666;font-size:80%;">18 August 2006</div><br />
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<p>The minimum number of cubes to cover every visible face on a cuboid measuring 3&nbsp;x&nbsp;2&nbsp;x&nbsp;1 is twenty-two.</p>
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<p>If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred and eighteen cubes to cover every visible face.</p>
<p>However, the first layer on a cuboid measuring 5&nbsp;x&nbsp;1&nbsp;x&nbsp;1 also requires twenty-two cubes; similarly the first layer on cuboids measuring 5&nbsp;x&nbsp;3&nbsp;x&nbsp;1, 7&nbsp;x&nbsp;2&nbsp;x&nbsp;1, and 11&nbsp;x&nbsp;1&nbsp;x&nbsp;1 all contain forty-six cubes.</p>
<p>We shall define C(<i>n</i>) to represent the number of cuboids that contain <i>n</i> cubes in one of its layers. So C(22) = 2, C(46) = 4, C(78) = 5, and C(118) = 8.</p>
<p>It turns out that 154 is the least value of <i>n</i> for which C(<i>n</i>) = 10.</p>
<p>Find the least value of <i>n</i> for which C(<i>n</i>) = 1000.</p>

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